Quasiparticle Wave Functions Research Articles

Effective mass of quasiparticles from thermodynamics

We discuss the potential advantages of calculating the effective mass of quasiparticles in the interacting electron liquid from the low-temperature free energy vis-a-vis the conventional approach, in which the effective mass is obtained from approximate calculations of the self-energy, or from a quantum Monte Carlo evaluation of the energy of a variational "quasiparticle wave function". While raw quantum Monte Carlo data are presently too sparse to allow for an accurate determination of the effective mass, the values estimated by this method are numerically close to the ones obtained in previous calculations using diagrammatic many-body theory. In contrast to this, a recently published parametrization of quantum Monte Carlo data for the free energy of the homogeneous electron liquid yields effective masses that considerably deviate from previous calculations and even change sign for low densities, reflecting an unphysical negative entropy. We suggest that this anomaly is related to the treatment of the exchange energy at finite temperature.

Axially deformed solution of the Skyrme–Hartree–Fock–Bogolyubov equations using the transformed harmonic oscillator basis (III) hfbtho (v3.00): A new version of the program

We describe the new version 3.00 of the code hfbtho that solves the nuclear Hartree–Fock (HF) or Hartree–Fock–Bogolyubov (HFB) problem by using the cylindrical transformed deformed harmonic oscillator basis. In the new version, we have implemented the following features: (i) the full Gogny force in both particle–hole and particle–particle channels, (ii) the calculation of the nuclear collective inertia at the perturbative cranking approximation, (iii) the calculation of fission fragment charge, mass and deformations based on the determination of the neck, (iv) the regularization of zero-range pairing forces, (v) the calculation of localization functions, (vi) a MPI interface for large-scale mass table calculations. PROGRAM SUMMARYProgram title:hfbtho v3.00Program Files doi:http://dx.doi.org/10.17632/c5g2f92by3.1Licensing provisions: GPL v3Programming language: FORTRAN-95Journal reference of previous version: M.V. Stoitsov, N. Schunck, M. Kortelainen, N. Michel, H. Nam, E. Olsen, J. Sarich, and S. Wild, Comput. Phys. Commun. 184 (2013).Does the new version supersede the previous one: YesSummary of revisions:1. the Gogny force in both particle–hole and particle–particle channels was implemented;2. the nuclear collective inertia at the perturbative cranking approximation was implemented;3. fission fragment charge, mass and deformations were implemented based on the determination of the position of the neck between nascent fragments;4. the regularization method of zero-range pairing forces was implemented;5. the localization functions of the HFB solution were implemented;6. a MPI interface for large-scale mass table calculations was implemented.Nature of problem:hfbtho is a physics computer code that is used to model the structure of the nucleus. It is an implementation of the energy density functional (EDF) approach to atomic nuclei, where the energy of the nucleus is obtained by integration over space of some phenomenological energy density, which is itself a functional of the neutron and proton intrinsic densities. In the present version of hfbtho, the energy density derives either from the zero-range Skyrme or the finite-range Gogny effective two-body interaction between nucleons. Nuclear super-fluidity is treated at the Hartree–Fock–Bogolyubov (HFB) approximation. Constraints on the nuclear shape allows probing the potential energy surface of the nucleus as needed e.g., for the description of shape isomers or fission. The implementation of a local scale transformation of the single-particle basis in which the HFB solutions are expanded provide a tool to properly compute the structure of weakly-bound nuclei.Solution method: The program uses the axial Transformed Harmonic Oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the Hartree–Fock–Bogolyubov Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions or the finite-range Gogny force until a self-consistent solution is found. A previous version of the program was presented in M.V. Stoitsov, N. Schunck, M. Kortelainen, N. Michel, H. Nam, E. Olsen, J. Sarich, and S. Wild, Comput. Phys. Commun. 184 (2013) 1592–1604 with much of the formalism presented in the original paper M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comput. Phys. Commun. 167 (2005) 43–63.Additional comments: The user must have access to (i) the LAPACK subroutines dsyeevr, dsyevd, dsytrf and dsytri, and their dependencies, which compute eigenvalues and eigenfunctions of real symmetric matrices, (ii) the LAPACK subroutines dgetri and dgetrf, which invert arbitrary real matrices, and (iii) the BLAS routines dcopy, dscal, dgemm and dgemv for double-precision linear algebra (or provide another set of subroutines that can perform such tasks). The BLAS and LAPACK subroutines can be obtained from the Netlib Repository at the University of Tennessee, Knoxville: http://netlib2.cs.utk.edu/.

Coordinate-space solver for superfluid many-fermion systems with the shifted conjugate-orthogonal conjugate-gradient method

Self-consistent approaches to superfluid many-fermion systems in 3-dimensions (and subsequent time-dependent approaches) require a large number of diagonalizations of very large dimension hermitian matrices, which results in enormous computational costs. We present an approach based on the shifted conjugate-orthogonal conjugate-gradient (COCG) method for the evaluation of the Green's function, from which we subsequently extract various densities (particle number, spin, current, kinetic energy, etc.) of a nuclear system needed in self-consistent approaches. The approach eschews the construction of the quasiparticle wavefunctions and their corresponding quasiparticle energies, which are never explicitly needed in any density functional approaches. As benchmarks we present calculations for nuclei with axial symmetry, including the ground state of spherical (magic or semi-magic) and axially deformed nuclei, the saddle-point in the $^{240}$Pu constrained fission path, and a vortex in the neutron star crust.

A simple derivation of the exact quasiparticle theory and its extension to arbitrary initial excited eigenstates.

The quasiparticle (QP) energies, which are minus of the energies required by removing or produced by adding one electron from/to the system, corresponding to the photoemission or inverse photoemission (PE/IPE) spectra, are determined together with the QP wave functions, which are not orthonormal and even not linearly independent but somewhat similar to the normal spin orbitals in the theory of the configuration interaction, by self-consistently solving the QP equation coupled with the equation for the self-energy. The electron density, kinetic, and all interaction energies can be calculated using the QP wave functions. We prove in a simple way that the PE/IPE spectroscopy and therefore this QP theory can be applied to an arbitrary initial excited eigenstate. In this proof, we show that the energy-dependence of the self-energy is not an essential difficulty, and the QP picture holds exactly if there is no relaxation mechanism in the system. The validity of the present theory for some initial excited eigenstates is tested using the one-shot GW approximation for several atoms and molecules.

New concept for the pairing anti-halo effect as a localized wave packet of quasiparticles

The pairing anti-halo effect is a phenomenon that a pairing correlation suppresses a divergence of nuclear radius, which happens for single-particle states with orbital angular momenta of $l$ = 0 and 1 in the limit of vanishing binding energy. While this effect has mainly been discussed in terms of Hartree-Fock-Bogoliubov (HFB) theory, we here use a three-body model and provide its new intuitive concept as a localized wave packet for a quasi-particle, that is, a coherent superposition of a weakly bound and continuum wave functions due to a pairing interaction. We show that the one-particle density in the three-body model can be directly expressed with such quasi-particle wave functions, which have a close analog to wave functions in the HFB approximation.

Effect of magnetic field on energy spectrum and localization of electron in CdS/HgS/CdS/HgS/CdS multilayered spherical nanostructure

The theoretical investigation of magnetic field effect on energy spectrum and localization of the electron and oscillator strengths of intraband quantum transitions in the nanostructure CdS/HgS/CdS/HgS/CdS is performed. The calculations are made in the framework of effective mass approximation and rectangular potential barriers model using the method of the expansion of quasi-particle wave functions over the complete basis of functions obtained as the exact solutions of the Schrodinger equation for the electron in the nanostructure without the magnetic field.It is shown that the magnetic field violates the spherical symmetry of the system and takes off the degeneration of energy spectrum with respect to the magnetic quantum number. The energy of the electron in the states with m≥0 increases when magnetic field enhances; for the states with m<0 these dependences are non-monotonous (decreasing at first and then increasing). Moreover, the ground state of electron is formed alternately by the states with m=0, −1, −2, …. Magnetic field influences on the distribution of quasi-particle density. It is shown that the electron significantly changes its localization in the nanostructure with two potential wells tunneling through the potential barrier under the effect of magnetic field, changing the oscillator strengths of intraband quantum transitions.

Hybrid functionals with fixed mixing parameter perform no better than PBE for fundamental band gaps of nanoscale materials

Hybrid functionals mixing the exact exchange with (semi)local functionals to reinstall the missing derivative discontinuity have been successfully employed to predict band gaps $({E}_{\mathrm{g}})$ in bulk semiconductors. Here we show that traditional hybrid functionals with fixed fractions of exact exchange do not perform significantly better than the most popular semilocal PBE-GGA functional for ${E}_{\mathrm{g}}$ of semiconductor nanostructures, since their band-gap corrections are essentially size independent. This is because they cannot respond properly to the variation in screening when size changes. They merely predict constant band-gap corrections to the PBE gaps in silicon nanowires (Si NWs) when wire diameter reduces, instead of the dramatic increase predicted by many-body $GW$ calculations. Moreover, these hybrid functionals generate almost identical wave functions compared with PBE for both bulk Si and Si NWs, whose overlaps with corresponding quasiparticle wave functions become much smaller than 1 for narrow NWs.

Correlation Effects in Scanning Tunneling Microscopy Images of Molecules Revealed by Quantum Monte Carlo.

Scanning tunneling microscopy (STM) and spectroscopy probe the local density of states of single molecules electrically insulated from the substrate. The experimental images, although usually interpreted in terms of single-particle molecular orbitals, are associated with quasiparticle wave functions dressed by the whole electron-electron interaction. Here we propose an ab initio approach based on quantum Monte Carlo to calculate the quasiparticle wave functions of molecules. Through the comparison between Monte Carlo wave functions and their uncorrelated Hartree-Fock counterparts we visualize the electronic correlation embedded in the simulated STM images, highlighting the many-body features that might be observed.

Compton profiles of NiO and TiO2 obtained from first principles GWA spectral function

In this work, we first use momentum density studies to understand strongly correlated electron behavior, which is typically seen in transition metal oxides. We observe that correlated electron behavior as seen in bulk NiO is due to the Fermi break located in the middle of overlapping spectral functions obtained from a GW (G is Green’s function and W is the screened Coulomb interaction) approximation (GWA) calculation while in the case of TiO2 we can see that the origin of the constant momentum distribution in lower momenta is due to a pile up of spectra before the Fermi energy. These observations are then used to compare our calculated Compton profiles with previous experimental studies of Fukamachi and Limandri. Our calculations for NiO are observed to follow the same trend as the experimental profile but it is seen to have a wide difference in the case of TiO2before the Fermi break. The ground state momentum densities differ significantly from the quasiparticle momentum density, thus stressing the importance of the quasiparticle wave function as the input for the study of charge density and the electron localization function. Finally we perform a calculation of the quasiparticle renormalization function, giving a quantitative description of the discontinuity of the GWA momentum density.

Identifying the mobility edges in a one-dimensional incommensurate model with p-wave superfluid

The mobility edges which separate the localized energy eigenstates from the extended ones exist normally only in three dimensional systems. For one-dimensional systems with random on-site potentials, one never encounters mobility edges, where all the eigenstates are localized. However, there are two kinds of 1D systems such as correlated disordered models, and the systems of exponentially decaying hopping kinetics, features of mobility edges at some specific values become possible. We study in this paper the properties of the mobility edges in a one-dimensional p-wave superfluid on an incommensurate lattice with exponentially decaying hopping kinetics. Without the p-wave superluid, the system displays a single mobility edge, which separates the extended regime from the localized one at a certain energy. Without the exponentially decaying hopping term, the system displays a phase transition from a topological superconductor to an Anderson localization at a certain disorder strength, where no mobility edge exists. We are interested in the influence of the p-wave superfluid on the mobility edge. By solving the Bogoliubov-de Gennes equation, the eigenvalues and the eigenfunctions are obtained. In order to identify the extending or the localized properties of the eigenvectors, we define an inverse participation ratio IPR. For an extended state, IPRn～1/L which goes to zero at a large L, and for a localized one, IPRn being constant. Therefore, the IPR can be taken as a criterion to distinguish the extended state from the localized one, while the mobility edge is defined as the boundary between two different states. We find that, with a p-wave superfluid, the system changes from a single mobility edge to a multiple one, and the number of mobility edges increases with the increased superfluid pairing order parameter. To further obtain the energy or the location of the mobility edge, we investigate the scaling behavior of wave functions by using a multifractal analysis, which is calculated through the scaling index . The minimum value of the index, with the values min= 1, 0min1, and min= 0, mean the extended, critical, and localized states, respectively. For the two consecutive states, the minima of the scaling index min when extrapolating to the large size limit between 0 and 1 signal the mobility edge. By exploring the corresponding Bogoliubov quasi-particle wave functions for the system under open boundary conditions together with the multifractal analysis for the system under periodic boundary conditions, we identify two mobility edges for the system of the p-wave superfluid pairing. Furthermore, we will investigate how the existence of the mobility edges influences the p-wave superfluid, and identify the phase diagram at the given parameters. We will in the future try to understand the relationship between the topological superfluid and the mobility edges.

Symmetry and topology of two-dimensional noncentrosymmetric superconductors

We present a detailed study of the gap symmetry and the quasiparticle wave function topology in two-dimensional superconductors without inversion center. The strong spin-orbit coupling of electrons with the crystal lattice makes it necessary to describe superconductivity in terms of one or more nondegenerate bands characterized by helicity. We develop a topological classification of the superconducting states using the integer-valued Maurer-Cartan invariants and the Bogoliubov Wilson loops, and also calculate the spectrum of fermionic boundary modes.

Orbital-resolved vortex-core states in FeSe superconductors: A calculation based on a three-orbital model

We study electronic structure of vortex core states of FeSe superconductors based on a t$_{2g}$ three-orbital model by solving the Bogoliubov-de Gennes(BdG) equation self-consistently. The orbital-resolved vortex core states of different pairing symmetries manifest themselves as distinguishable structures due to different quasi-particle wavefunctions. The obtained vortices are classified in terms of the invariant subgroups of the symmetry group of the mean-field Hamiltonian in the presence of magnetic field. Isotropic $s$ and anisotropic $s$ wave vortices have $G_5$ symmetry for each orbital, whereas $d_{x^2-y^2}$ wave vortices show $G^{*}_{6}$ symmetry for $d_{xz/yz}$ orbitals and $G^{*}_{5}$ symmetry for $d_{xy}$ orbital. In the case of $d_{x^2-y^2}$ wave vortices, hybridized-pairing between $d_{xz}$ and $d_{yz}$ orbitals gives rise to a relative phase difference in terms of gauge transformed pairing order parameters between $d_{xz/yz}$ and $d_{xy}$ orbitals, which is essentially caused by a transformation of co-representation of $G^{*}_{5}$ and $G^{*}_{6}$ subgroup. The calculated local density of states(LDOS) of $d_{x^2-y^2}$ wave vortices show qualitatively similar pattern with experiment results. The phase difference of $\frac{\pi}{4}$ between $d_{xz/yz}$ and $d_{xy}$ orbital-resolved $d_{x^2-y^2}$ wave vortices can be verified by further experiment observation.

Nambu-Goldstone modes propagating along topological defects: Kelvin and ripple modes from small to large systems

Nambu-Goldstone modes associated with (topological) defects such as vortices and domain walls in (super)fluids are known to possess quadratic/non-integer dispersion relations in finite/infinite-size systems. Here, we report interpolating formulas connecting the dispersion relations in finite- and infinite-size systems for Kelvin modes along a quantum vortex and ripplons on a domain wall in superfluids. Our method can provide not only the dispersion relations but also the explicit forms of quasiparticle wavefunctions $(u,v)$. We find a complete agreement between the analytical formulas and numerical simulations. All these formulas are derived in a fully analytical way, and hence not empirical ones. We also discuss common structures in the derivation of these formulas and speculate on the general procedure.

Excitation spectra and wave functions of quasiparticle bound states in bilayer Rashba superconductors

We study the excitation spectra and the wave functions of quasiparticle bound states at a vortex and an edge in bilayer Rashba superconductors under a magnetic field. In particular, we focus on the quasiparticle states at the zero energy in the pair-density wave state in a topologically non-trivial phase. We numerically demonstrate that the quasiparticle wave functions with zero energy are localized at both the edge and the vortex core if the magnetic field exceeds the critical value.

Convergence of quasiparticle self-consistentGWcalculations of transition-metal monoxides

Finding an accurate ab initio approach for calculating the electronic properties of transition-metal oxides has been a problem for several decades. In this paper, we investigate the electronic structure of the transition-metal monoxides MnO, CoO, and NiO in their undistorted rocksalt structure within a fully iterated quasiparticle self-consistent $GW$ ($\mathrm{QPsc}GW$) scheme. We study the convergence of the $\mathrm{QPsc}GW$ method, i.e., how the quasiparticle energy eigenvalues and wave functions converge as a function of the $\mathrm{QPsc}GW$ iterations, and we compare the converged outputs obtained from different starting wave functions. We find that the convergence is slow and that a one-shot ${G}_{0}{W}_{0}$ calculation does not significantly improve the initial eigenvalues and states. It is important to notice that in some cases the ``path'' to convergence may go through energy band reordering which cannot be captured by the simple initial unperturbed Hamiltonian. When we reach a fully iterated solution, the converged density of states, band gaps, and magnetic moments of these oxides are found to be only weakly dependent on the choice of the starting wave functions and in reasonably good agreement with the experiment. Finally, this approach provides a clear picture of the interplay between the various orbitals near the Fermi level of these simple transition-metal monoxides. The results of these accurate ab initio calculations can provide input for models aiming at describing the low-energy physics in these materials.

Non-Gaussian spatial correlations dramatically weaken localization.

We perform variational studies of the interaction-localization problem to describe the interaction-induced renormalizations of the effective (screened) random potential seen by quasiparticles. Here we present results of careful finite-size scaling studies for the conductance of disordered Hubbard chains at half-filling and zero temperature. While our results indicate that quasiparticle wave functions remain exponentially localized even in the presence of moderate to strong repulsive interactions, we show that interactions produce a strong decrease of the characteristic conductance scale g^{*} signaling the crossover to strong localization. This effect, which cannot be captured by a simple renormalization of the disorder strength, instead reflects a peculiar non-Gaussian form of the spatial correlations of the screened disordered potential, a hitherto neglected mechanism to dramatically reduce the impact of Anderson localization (interference) effects.

Improved quasiparticle wave functions and mean field forG0W0calculations: Initialization with the COHSEX operator

The GW approximation to the electron self-energy has become a standard method for ab initio calculation of excited-state properties of condensed-matter systems. In many calculations, the G W self-energy operator, E, is taken to be diagonal in the density functional theory (DFT) Kohn-Sham basis within the G0 W0 scheme. However, there are known situations in which this diagonal Go Wo approximation starting from DFT is inadequate. We present two schemes to resolve such problems. The first, which we called sc-COHSEX-PG W, involves construction of an improved mean field using the static limit of GW, known as COHSEX (Coulomb hole and screened exchange), which is significantly simpler to treat than GW W. In this scheme, frequency-dependent self energy E(N), is constructed and taken to be diagonal in the COHSEX orbitals after the system is solved self-consistently within this formalism. The second method is called off diagonal-COHSEX G W (od-COHSEX-PG W). In this method, one does not self-consistently change the mean-field starting point but diagonalizes the COHSEX Hamiltonian within the Kohn-Sham basis to obtain quasiparticle wave functions and uses the resulting orbitals to construct the G W E in the diagonal form. We apply both methods to a molecular system, silane, and to two bulk systems, Si and Ge under pressure. For silane, both methods give good quasiparticle wave functions and energies. Both methods give good band gaps for bulk silicon and maintain good agreement with experiment. Further, the sc-COHSEX-PGW method solves the qualitatively incorrect DFT mean-field starting point (having a band overlap) in bulk Ge under pressure.

Quasiclassical Treatment and Odd-Parity/Triplet Correspondence in Topological Superconductors

We construct a quasiclassical framework for topological superconductors with the strong spin-orbit coupling such as CuxBi2Se3. In the manner of the quasiclassical treatment, decomposing the slowly varying component from the total quasi-particle wave function, the original massive Dirac Bogoliubov-de Gennes (BdG) Hamiltonian derived from the tight-binding model represented by 8 x 8 matrix is reduced to 4 x 4 one. The resultant equations are equivalent to Andreev-type equations of singlet or triplet superconductors, in which the apparent spin-orbit coupling vanishes. Using this formalism, we find a fact that the odd-parity superconductivity in topological superconductors turns to the spin-triplet one. % without the spin-orbit coupling through the quasiclassical treatment. Moreover, in terms of the quasiclassical treatment, we show that the topologically-protected zero-energy states in topological superconductors has the correspondence to the Andreev bound states established in a long history of studies for the unconventional superconductors. This clearly indicates that low-energy non-trivial superconducting properties in the topological superconductors can be analyzed by the established theoretical descriptions on the spin-triplet superconductors.

On Gravitational Effects in the Schrödinger Equation

The Schrodinger equation for a particle of rest mass \(m\) and electrical charge \(ne\) interacting with a four-vector potential \(A_i\) can be derived as the non-relativistic limit of the Klein–Gordon equation \(\left( \Box '+m^2\right) \varPsi =0\) for the wave function \(\varPsi \), where \(\Box '=\eta ^{jk}\partial '_j\partial '_k\) and \(\partial '_j=\partial _j -\mathrm {i}n e A_j\), or equivalently from the one-dimensional action \(S_1=-\int m ds +\int neA_i dx^i\) for the corresponding point particle in the semi-classical approximation \(\varPsi \sim \exp {(\mathrm {i}S_1)}\), both methods yielding the equation \(\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi \) in Minkowski space–time , where \(\alpha ,\beta =1,2,3\) and \(\phi =-A_0\). We show that these two methods generally yield equations that differ in a curved background space–time \(g_{ij}\), although they coincide when \(g_{0\alpha }=0\) if \(m\) is replaced by the effective mass \(\mathcal{M}\equiv \sqrt{m^2-\xi R}\) in both the Klein–Gordon action \(S\) and \(S_1\), allowing for non-minimal coupling to the gravitational field, where \(R\) is the Ricci scalar and \(\xi \) is a constant. In this case \(\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi \), where \(\phi ^{(\mathrm g)} =\sqrt{g_{00}}\) and \(\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} \), the correctness of the gravitational contribution to the potential having been verified to linear order \(m\phi ^{(\mathrm g)} \) in the thermal-neutron beam interferometry experiment due to Colella et al. Setting \(n=2\) and regarding \(\varPsi \) as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div\({{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0\), where \({{\varvec{A}}}^{\alpha }=-A^{\alpha }\) and \({{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }\). The quantum-cosmological Schrodinger (Wheeler–DeWitt) equation is also discussed in the \(\mathcal{D}\)-dimensional mini-superspace idealization, with particular regard to the vacuum potential \(\mathcal V\) and the characteristics of the ground state, assuming a gravitational Lagrangian \(L_\mathcal{D}\) which contains higher-derivative terms up to order \(\mathcal{R}^4\). For the heterotic superstring theory , \(L_\mathcal{D}\) consists of an infinite series in \(\alpha '\mathcal{R}\), where \(\alpha '\) is the Regge slope parameter, and in the perturbative approximation \(\alpha '|\mathcal{R}| \ll 1\), \(\mathcal V\) is positive semi-definite for \(\mathcal{D} \ge 4\). The maximally symmetric ground state satisfying the field equations is Minkowski space for \(3\le {\mathcal {D}}\le 7\) and anti-de Sitter space for \(8 \le \mathcal {D} \le 10\).

Gradient of molecular Hartree–Fock–Bogoliubov energy with a linear combination of atomic orbital quasiparticle wave functions

The analytical gradient for the atomic-orbital-based Hartree-Fock-Bogoliubov (HFB) energy functional, the modified form of which was proposed by Staroverov and Scuseria to account for the static electron correlation [J. Chem. Phys. 117, 11107 (2002)], is derived. Interestingly, the Pulay force for the HFB energy is expressed with the same formula as that for the Hartree-Fock method. The efficiency of the present HFB energy gradient is demonstrated in the geometry optimizations of conjugated and biradical systems. The geometries optimized by using the HFB method with the appropriate factor ζ, which controls the degree of static correlation included, are found to show good agreement with those obtained by using a complete active-space self-consistent field method, although they are significantly dependent on ζ.